Rajan, whose tenure as the RBI chief will end on September 4, had on last Saturday said he will not be seeking an extension as the governor.
The home currency opened sharply lower at 67.65 per dollar as against the last weekend's level of 67.08 at the Interbank Foreign Exchange (Forex) Market and dropped further to one-month low of 67.70 on initial heavy dollar demand.
The domestic currency had ended higher by 13 paise or 0.19 per cent on last Friday.
The local currency hovered in a range of 67.28 and 67.70 per dollar during the day.
Meanwhile, the dollar index was down by 0.48 per cent at against the basket of six global currencies in the late afternoon trade.
In cross-currency trades, the rupee fell further against the pound sterling to close at 98.49 from 95.86 on last Friday and also moved down against the euro to 76.25 from 75.53.
However, the domestic currency inched up further against the Japanese yen to 64.37 per 100 yens from 64.38.
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